Problem: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 16,\enspace 11,\enspace 10,\enspace 2,\enspace 10$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{16 + 11 + 10 + 2 + 10}{{5}} = {9.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $16$ years $6.2$ years $38.44$ years $^2$ $11$ years $1.2$ years $1.44$ years $^2$ $10$ years $0.2$ years $0.04$ years $^2$ $2$ years $-7.8$ years $60.84$ years $^2$ $10$ years $0.2$ years $0.04$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{38.44} + {1.44} + {0.04} + {60.84} + {0.04}} {{5}} $ $ {\sigma^2} = \dfrac{{100.8}}{{5}} = {20.16\text{ years}^2} $ The average sloth at the zoo is 9.8 years old. The population variance is 20.16 years $^2$.